A careers advisor by trade, Amesbury previously worked as a manager with the Connexions careers advice service. He also worked as Senior Parliamentary Advisor to Shadow Education Secretary Angela Rayner and as political advisor and stakeholder manager on Andy Burnham's successful campaign to become Metro Mayor for Greater Manchester.[7] He was a director of City South Manchester Housing Trust, an award-winning social enterprise providing affordable housing in the Fallowfield, Hulme, Moss Side and Whalley Range areas of south Manchester. He also served as a policy advisor to Tameside Council.

Amesbury joined the Labour Party as a 17-year-old after moving with his family to Yorkshire, where he says he was politicised by the impact on local families of Margaret Thatcher's political policies and the subsequent miners' strike.[8] He served as a Regional Officer and Fundraising and Events Manager for Labour and was later elected to the party's National Policy Forum (NPF)—the policy making arm of The Labour Party. He became a councillor on Manchester City Council representing Fallowfield from 2006, rising to the position of Executive Cabinet Member for Culture and Leisure where he helped bring the National Football Museum to the city, standing down when he was elected to the House of Commons.[9][10]

He gained the Weaver Vale constituency from the Conservative incumbent Graham Evans in the 2017 general election on a swing of 4.3%. He was named Parliamentary Private Secretary in the Shadow Department for Work and Pensions team in January 2018, and in July of the same year was promoted to Shadow Employment Minister.[11][12]

In 2019, Amesbury apologised for having shared an antisemitic caricature on Facebook in 2013, stating “I apologise unreservedly for this terrible error. I genuinely don’t recall sharing this image and I’m mortified that I did so. This appalling image image contains an antisemitic caricature and a reference to the ‘illuminati’ conspiracy theory. I would never have intentionally shared antisemitic tropes and I am sincerely sorry that I did.” [13]